Solid Propellant Rocket Research [1 ed.] 9781600862571, 9781563478505

Citation preview

Purchased from American Institute of Aeronautics and Astronautics

Solid Propellant Rocket Research

Purchased from American Institute of Aeronautics and Astronautics

Progress in ASTRONAUTICS and ROCKETRY A series of volumes sponsored by American Rocket Society 500 Fifth Avenue, New York 36, New York

Progress Series Editor Martin Summerfield Princeton University, Princeton, New Jersey

Titles in the Series Volume 1. Solid Propellant Rocket Research. 1960 Editor: MARTIN SUMMERFIELD In preparation Volume 2. Liquid Rockets and Propellants. 1960 Editors: LOREN E. BOLLINGER, MARTIN GOLDSMITH, AND ALEXIS W. LEMMON, JR. Volumes 3 and 4. Space Power Systems. 1961 Editor: NATHAN W. SNYDER Volume 5. Electrostatic Propulsion. 1961 Editors: DAVID B. LANGMUIR, ERNST STUHLINGER, AND J. M. SELLEN (Other volumes are planned)

ACADEMIC PRESS • NEW YORK AND LONDON

Purchased from American Institute of Aeronautics and Astronautics

Propellent Rocket Research Edited by

Martin Summerfield Princeton University, Princeton, New Jersey

A Selection of Technical Papers based mainly on A Symposium of the American Rocket Society held at Princeton University, Princeton, New Jersey, January 28, 29, 1960.

ACADEMIC PRESS • NEW YORK • LONDON • 1960

Purchased from American Institute of Aeronautics and Astronautics

COPYRIGHT© I960, BY ACADEMIC PRESS INC. ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS INC. Ill FIFTH AVENUE NEW YORK 3, N. Y.

United Kingdom Edition Published by ACADEMIC PRESS INC. (London) LTD. 17 OLD QUEEN STREET, LONDON S. W. 1

Library of Congress Catalog Card Number 60-16913

PRINTED IN THE UNITED STATES OF AMERICA

Purchased from American Institute of Aeronautics and Astronautics

AMERICAN ROCKET SOCIETY

Solid Rockets Committee January 1960 Ivan E. Tuhy, Chairman Rocketdyne, a Division of North American Aviation, Inc .

William C. Fagan Wright Air Development Center, U . S . Air Force Fred K. Guest Space Technology Laboratories, Inc . A. Terrell Jones Army Ballistic Missile Agency H. Griffeth Jones Elkton Division, Thiokol Chemical Corp. Emil A. Malick Phillips Petroleum Co . John R. McDonald University of Houston Arch C. Scurlock Atlantic Research Corp . Edward J. Sheehy Allegany Ballistics Laboratory Robert Sherman U . S . Military Academy Irwin A. Spitzer Grand Central Rocket Co . David F. Sprenger Aerojet-General Corp.

Symposium Papers Committee Martin Summerfield Princeton University Ivan E. Tuhy Rocketdyne, a Division of North American Aviation, Inc.

Purchased from American Institute of Aeronautics and Astronautics

This page intentionally left blank

Purchased from American Institute of Aeronautics and Astronautics

CONTRIBUTORS TO VOLUME 1

Andersen, W.H., Ordnance Division, Aerojet-General Corporation, Azusa, California.. .p. 227.

Angelus, T.A., Allegany Ballistics Laboratory, Hercules Powder Company, Cumberland, Maryland. .. p. 527.

Au, N.N., Aero/Space Vehicles Laboratory, Hughes Aircraft Company, Culver City, California. . .p. 101. Baer, A. D,, Department of Chemical Engineering, University of Utah, Salt Lake City, Utah. . .p. 653. Bastress, E.K., Department of Aeronautical Engineering, Princeton University, Princeton, New Jersey • P-183. Beyer, Rodney B., Chemistry Department, Stanford Research Institute, Menlo Park, California, .p. 673. Bird, J.F., Applied Physics Laboratory, The Johns Hopkins University, Silver Spring, Maryland. . .pp.295, 423.

Blair, D. W., Department of Aeronautical Engineering, Princeton University, Princeton, New Jersey, p. 183. Boyd, A.B., Solid Propulsion Operations, Rocketdyne, a Division of North American Aviation, Inc., McGregor, Texas... p. 3. Brownlee, W. Grant, Daniel and Florence Guggenheim Jet Propulsion Center, California Institute of Technology, Pasadena, California. .. p. 455. Burkes, W. M., Solid Propulsion Operations, Rocketdyne, a Division of North American Aviation, Inc., McGregor, Texas . . . p. 3.

Chaiken, R.F., Chemical Division, Aerojet-General Corporation, Azusa, California. . .p. 227. Cheng, Sin-I, Department of Aeronautical Engineering, Princeton University, Princeton, New Jersey, p. 393.

Cowan, P. L., Stanford Research Institute, Palo Alto, California. . .p. 623. Vll

Purchased from American Institute of Aeronautics and Astronautics

Dishon, Menachem, Ordnance Corps, Israel Defense Forces, Tel Aviv, Israel... p. 359.

Fassell, W. M., Materials Department, Aeronutronic Division of Ford Motor Company, Newport Beach, California ... p. 259. Fishman, Norman, Chemistry Department, Stanford Research Institute, Menlo Park, California, p. 673.

Freudenthal, A . M . , Department of Civil Engineering, Columbia University, New York, New York. p. 33. Classman, Irvin, Department of Aeronautical Engineering, Princeton University, Princeton, New Jersey, p. 253.

Gordon, Derck A., Propulsion Division, Poulter Laboratory, Stanford Research Institute, Menlo Park, California. . .p. 271. Hall, K. P. , Department of Aeronautical Engineering, Princeton University, Princeton, New Jersey. . .pp. 141, 183.

Hart, R . W . , Applied Physics Laboratory, The Johns Hopkins University, Silver Spring, Maryland. .. pp. 295,423. Henry, L. A., Department of Civil Engineering, Columbia University, New York, New Y o r k . . . p. 33.

Hermance, C . E . , Department of Aeronautical Engineering, Princeton University, Princeton, New Jersey, p. 183. Hildenbrand, D. L., Materials Department, Aeronutronic Division of Ford Motor Company, Newport Beach, California. . . p. 259. Kuby, W . C . , Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California. .. p. 495. Lampens, G., Centre de Recherches pour Tlndustrie des Produits Explosifs, Poudreries Reunies de Belgique, Brussels, Belgium... p. 121.

Landsbaum, E . M . , Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California . p. 495. Marble, Frank E . , Daniel and Florence Guggenheim Jet Propulsion Center, California Institute of Technology, Pasadena, California ... p. 455.

McAlevy, R . F . , in, Department of Aeronautical Engineering, Princeton University, Princeton, New Jersey, p. 623. McClure, F.T. , Applied Physics Laboratory, The Johns Hopkins University, Silver Spring, Maryland. . .pp. 295, 423.

Purchased from American Institute of Aeronautics and Astronautics

Medford, J . E . , Solid Propulsion Operations, Rocketdyne, a Division of North American Aviation, Inc., McGregor, Texas . . .p. 3. Nachbar, W,, Lockheed Missiles and Space Division, Sunny vale, California... p. 207. Papp, C.A., Materials Department, Aeronutronic Division of Ford Motor Company, Newport Beach, California. . .p. 259. Price, E . W . , U.S. Naval Ordnance Test Station, China Lake, California. . .p. 561.

Ryan, N . W . , Department of Chemical Engineering, University of Utah, Salt Lake City, Utah. . .p. 653. Salt, D. L., Department of Chemical Engineering, University of Utah, Salt Lake City, Utah. . .p. 653.

Sernka, R. P., Materials Department, Aeronutronic Division of Ford Motor Company, Newport Beach, California. . .p. 259. Shinnar, Reuel, Technion, Israel Institute of Technology, Haifa, Israel... p. 359. Smith, A. G., Department of Aircraft Propulsion, The College of Aeronautics, Cranfield, Bletchley, Bucks., England. . .p. 375. Spaid, F.W., Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California. . .p. 495.

Summerfield, Martin, Department of Aeronautical Engineering, Princeton University, Princeton, New Jersey .. .pp. 141, 183, 623. Sutherland, G.S., Systems Management Office, Boeing Airplane Company, Seattle, Washington .. .p. 141. Taback, H.J., Applied Mechanics Division, Aerojet-General Corporation, Sacramento, California. .. p. 141. Talley, Claude P., Experiment Incorporated, A subsidiary of Texaco Inc., Richmond, Virginia. . .p. 279.

Vandenkerckhove, J., Institute of Aeronautics, University of Brussels, Brussels, Belgium. . . p . 121. Wall, Richard H., Redstone Division, Thiokol Chemical Corporation, Huntsville, Alabama... p. 603. Webb, M.J., Department of Aeronautical Engineering, Princeton University, Princeton, New Jersey .p. 141.

Purchased from American Institute of Aeronautics and Astronautics

Williams, M. L., Guggenheim Aeronautical Laboratory, California Institute of Technology, Pasadena, California. . .p. 67. Wood, William A., Combustion Group, Rohm and Haas Company, Hunts ville, Alabama... p. 287.

Purchased from American Institute of Aeronautics and Astronautics

PREFACE TO THE SERIES

This volume on Solid Propellant Rocket Research is the first in a new series, Progress in Astronautics and Rocketry, sponsored by the American Rocket Society. The initiation of this new project by the ARS was prompted mainly by the recognition that, in recent times, the offerings of worthy technical manuscripts to the Society for publication have exceeded by far the capacity of its two existing publications, ARS Journal and Astronautics, to publish them and, further, that this publishing gap threatened to become worse in the future. In particular, a severe load is expected as a result of the increasing number of specialized symposia in various fields that are being scheduled by the twenty-one Technical Committees of the Society. This new series represents, however, more than a mere quantitative solution to the problem of publishing a large number of papers. It is a new approach to the basic task of publishing the scientific output of a society as diverse in its specialties as the ARS. Instead of aiming at a broad service to the whole field in each issue, as the ARS Journal does and as most other periodicals do, each volume of this series will be focused on a single specialty, generally the areaof a particular Technical Committee. There is reason to believe that this approach will enhance the usefulness of each volume and lead to effective distribution of the papers thus published. At the same time, the ARS Journal will be liberated, to a great extent, from its severe backlog problem, and it will be able to deal more effectively with its proper assignment of publishing promptly individual research papers of merit. The ARS wishes to record here its appreciation of the spirit of cooperation demonstrated by Academic Press Inc., in accepting the publishing responsibility for the Series.

September, 1960 Martin Summerfield Series Editor

Purchased from American Institute of Aeronautics and Astronautics

This page intentionally left blank

Purchased from American Institute of Aeronautics and Astronautics

PREFACE TO THE VOLUME

The present volume is an outgrowth of a symposium on Solid Propellant Rocket Research held at Princeton University on January 28 and 29, 1960, sponsored jointly by the American Rocket Society and the Conference Organization of the University. Although most of the twenty-seven papers in this volume were actually presentedat the symposium, sixwere drawn from the backlog of unpublished manuscripts awaiting publication in the Society1 s ARS Journal, and several are specially-requested, much-expanded revisions of the papers that were presented. In selecting the papers for the volume, the editor attempted to meet as nearly as possible the present needs of solid rocket research scientists for an up-to-date view of the most promising ideas in certain special areas of researcho Most of the book deals with the combustion of solid propellants. The particular topics singled out for attention are: Steady-state burning of composite propellants, combustion of metals, unstable burning in solid propellant rockets, and ignition. In addition, the book includes a section on mechanical properties of solid propellant grains, essentially as a subject in its own right, but partly for its practical connection with internal ballistic design and hence with combustion. Many topics in solid propellant research are not represented at all--thermochemistry of propellants, chemical kinetics of propellant gases, nozzle design, heat transfer, etc. It was simply not possible to do the whole job in one book or in one symposium.

The papers chosen for this book were selected largely for their orientation toward scientific under standing of basic phenomena, although several papers on the engineering aspects of a topic were also included in order to provide the technical background of the problem on which the research papers are centered. In each topical area covered, this book brings together in one place for the first time some of the most significant lines of current research. Thus, the reader T s attention is directed particularly to the papers on unstable burning, both theoretical and experimental. In the theoretical papers, diverse treatments are presented for the basic mechanism of interaction between a small-

Purchased from American Institute of Aeronautics and Astronautics

amplitude, acoustic-type oscillation in the gas-filled chamber of a rocket motor and the flame zone at the surface of the burning propellant. The question of stability or instability of the oscillation depends on the nature of this interaction and, consequently, on the structure of the thin flame zone.

The question of the structure of the thin flame zone is important not only in the treatment of unstable burning but, of course, in the steady-state burning-rate field as well. In the section on steady-state burning mechanisms, three different approaches are offered to the analysis of the factors that control the burning of a composite-type solid propellant. In these papers, three physical models are developed for the burning process, one based on a granular diffusion flame, a second on a two-dimensional diffusion flame, and a third on the proposition that the ammonium perchlorate decomposition flame dominates the combustion process. An intriguing aspect of the combustion of a solid propellant is the influence on its burning properties of minor percentages of powdered metal incorporated in the propellant, particularly the interaction of the propellant flame with gas-dynamic oscillations in the combustion chamber. Some novel researches on the mechanism of burning of metals and the rates of burning are described in a group of five papers in this volume. The ignition process of a solid propellant is an interesting example of flame physics with important practical applications. The three papers on this subject deal with the basic question of the site of the first flame, whether it occurs in the gas phase or in the solid phase, and the processes that lead to this initialflame. Depending on the conclusions that will ultimately emerge from these studies, it will be possible to predict, on the basis of fundamental principles, the ignitability of a particular propellant and the igniting capability of a particular ignition system. The section on mechanical properties deals with the vital problem of calculating the strain distribution in solid propellant grains under the influence of internal pressure with various conditions of restraint or under the influence of temperature gradients. A complicating aspect of the problem is that a solid propellant is a peculiar viscoelastic medium, and not a perfectly elastic solid, thereby obscuring to a great degree the criterion for ultimate failure of agrain. This is a subject of great practical importance. The mechanical properties of the solid propellant grain, particularly its acoustic properties, are pertinent also to the prediction of combustion instability, as brought out in one of the papers in the theoretical instability section.

Purchased from American Institute of Aeronautics and Astronautics

The contributions in this volume, collected as a group, will undoubtedly serve to stimulate further research in these subjects . Although it is an essential limitation of this kind of collection that it cannot provide total coverage of a subject, an attempt was made to supply, in each paper, the important references thatwould enable the reader to explore more completely the area of his interest. September, 1960

Martin Summerfield Volume Editor

Purchased from American Institute of Aeronautics and Astronautics

This page intentionally left blank

Purchased from American Institute of Aeronautics and Astronautics

CONTENTS Solid Rockets Committee and Symposium Papers Committee . . . . . . . . . . Contributors to Volume 1 . . . . . . . .

Preface to the Series . . . . . . . . . . Preface to the Volume . . . . . . . . . . A. Mechanical Properties of Grains Grain Design and Development Problems for Very Large Rocket Motors . . . . . . . . . . A.B.Boyd, W.M. Burkes and J.E.Medford On *PoissonTs Ratio " in Linear Visco-Elastic Propellants . . . . . . . . . . . . . . . . . . .

3

33

A. M. Freudenthal and L. A. Henry

Mechanical Properties and the Design of Solid Propellant Motors . . . . . . . . . . . . . . M.L. Williams

67

A Method of Strength Analysis of Solid Propellant Rocket Grains . . . . . . . . . . . . N.N.Au

101

Stress and Strain Analysis of Cylindrical Case-Bonded Grains . . . . . . . . . . . . . . . . J. Vandenkerckhove and G. Lampens

121

B. Steady-State Burning Mechanisms

Burning Mechanism of Ammonium Perchlorate Propellants . . . . . . . . . . . . . . . . . . . . Martin Summerfield, G.S.Sutherland, M.J. Webb, H. J. Taback and K. P. Hall

141

Some Research Problems in the Steady-State Burning of Composite Solid Propellants . . . . . D.W.Blair, E.K.Bastress, C.E.Hermance, K. P. Hall and M. Summerfield

183

A Theoretical Study of the Burning of a Solid Propellant Sandwich . . . . . . . . . . . . . W. Nachbar

207

xvii

Purchased from American Institute of Aeronautics and Astronautics

The Role of Binder in Composite Propellant Combustion . . . . . . . . . . . . . . . R. F. Chaiken and W. H. Andersen

227

C. Combustion of Metals Combustion of Metals: Physical Considerations. . . Irvin Glassman

253

The Experimental Nature of the Combustion of Metallic Powders . . . . . . . . . . . . . . . . W. Mo Fassell, C.A.Papp, D. L.Hildenbrand and R. P. Sernka Combustion Characteristics of Metal Particles . . . . . . . . . . . . . . . . . . . . . Derck A. Gordon

The Combustion of Elemental Boron. . . . . . . . . Claude P.Talley Metal Combustion in Deflagrating Propellant . . . . . . . . . . . . . . . . . . . . . William A. Wood

. .

259

271 279

287

D. Theories of Unstable Combustion Solid Propellant Rocket Motors as Acoustic Oscillators . . . . . . . . . . . . . . . . . . . . . F.T.McClure, R. W. Hart and J. F. Bird Heat Transfer Stability Analysis of Solid Propellant Rocket Motors . . . . . . . . . . Reuel Shinnar and Menachem Dishon A Theory of Oscillatory Burning of Solid Propellants Assuming a Constant Surface Temperature . . . . . . . . . . . . • • A. G. Smith Combustion Instability in Solid Rockets Using Propellants with Reactive Additives . . . . . . . . . . . . . . . . . . . . . Sin-I Cheng

The Influence of Erosive Burning on Acoustic Instability in Solid Propellant Rocket Motors . . . . . . . . . . . . . . . . . . . . R.W.Hart, J.F.Bird and F.T.McClure

xviii

. .

295

359

375

393

423

Purchased from American Institute of Aeronautics and Astronautics

E. Experiments on Unstable Burning

An Experimental Investigation of Unstable Combustion in Solid propellant Rocket Motors . . . . . . . . . . . . . W. Grant Brownlee and Frank E. Marble Experimental Investigations of Unstable Burning in Solid Propellant Rocket Motors . . . . . . . . . . . . . . . . . . . . . . E. M. Landsbaum, W. C. Kuby and F. W. Spaid

Unstable Burning Phenomenon in Double-Base Propellants . . . . . . . . . . . . . . . . . . . . . Theos A.Angelus Review of Experimental Research on Combustion Instability of Solid Propellants . . . . . . . . . . . . . . . . . . . . . E.W. Price

Resonant Burning of Solid Propellants: Review of Causes, Cures and Effects . . . . . . . R.H.Wall

455

495

527

561

603

F. Solid Propellant Ignition The Mechanism of Ignition of Composite Solid Propellants by Hot Gases. . . . . . . . . . . R. F. McAlevy, III, P. L. Cowan and M. Summerfield Propellant Ignition by High Convective Heat Fluxes . . . . . . . . . . . . . . . . . . . . . A.D.Baer, N.W.Ryan and D. L. Salt

Solid Propellant Ignition Studies With High Flux Radiant Energy as a Thermal Source. . . . . . . . . . . . . . . . . . . Rodney B.Beyer and Norman Fishman

623

653

673

Purchased from American Institute of Aeronautics and Astronautics

This page intentionally left blank

Purchased from American Institute of Aeronautics and Astronautics

This page intentionally left blank

Purchased from American Institute of Aeronautics and Astronautics

SOLID PROPELLANT ROCKET RESEARCH

GRAIN DESIGN AND DEVELOPMENT PROBLEMS FOR VERY LARGE ROCKET MOTORS* A. B. Boyd, W. M. Burkes and J. E. Medford Solid Propulsion Operations Rocketdyne, a Division of North American Aviation, Inc. McGregor, Texas ABSTRACT This paper presents various approaches to grain design and development of very large solid propellant rocket motors, and introduces characteristic problems resulting from the size, weight, transportation limitations and propellant physical properties. To illustrate the problems in large solid propellant engines and an approach to their analysis, the design of a two million pound thrust motor using 720,000 pounds of propellant with typical physical properties was selected to give realistic quantitative solutions. Since transportation of a complete unit of this size is impractical, two approaches are considered: a single grain design utilizing on-site casting methods, and a modular design using multiple portable grains which are assembled at the launching site. INTRODUCTION This paper presents various approaches to the design of very large solid propellant rocket motors. The approaches presented are the monolithic, or single grain design, the compartmented grain, and the modular grain. The characteristic problems in these motors are discussed, and the applicability of the various approaches to the solution of these problems is outlined. The designs incorporating a minimum of propellant support devices are presented first, and these are followed by the other approaches in the order of increasing numbers of supports utilized. * Presented at ARS Solid Propellant Rocket Conference, Princeton, New Jersey, January 28-29, 1960.

Purchased from American Institute of Aeronautics and Astronautics

SOLID PROPELLANT ROCKET RESEARCH

To illustrate the problems and an approach to their analysis, the design of a two million pound thrust motor using 720,000 pounds of propellant with typical mechanical properties was selected for analysis. This motor has a diameter of 15 feet, an overall length of 66 feet and a grain length of 53 feet. Table I presents a summary of the design parameters for the three approaches. Figure 1 is an artist's conception of this motor in final assembly as the first stage of a rocket propelled vehicle. The monolithic design consists of a single propellant grain, bonded to the case wall. The propellant is cast directly into a single, continuous cavity in the case.

The grain in the compartmented design is divided into several sections separated by inert sheets or webs mounted in the case. Casting is done directly into the motor, but the grain does not form a continuous mass. In the modular design several propellant modules are cast and cured separately and are later assembled into the motor case. Propellant processing for very large rocket motors poses no unusual problems, therefore the paper is directed at other more critical problem areas. MOTOR HARDWARE

Although the main emphasis of this paper is on the grain design, a few remarks relative to the motor hardware are in order. The motor case which has been assumed for all three approaches consists of a wound glass fiber, reinforced plastic cylindrical section with a metal flange at the aft end which connects by means of tension bolts to a formed metal aft head. The forward head can be of formed metal construction and attached to the cylindrical section in the same manner as the

aft head, or it can be an integral part of the plastic cylindrical section. The case will be helically wound on-site for the motor considered and for any other motor 15 feet in diameter or larger. This is done in order to avoid transportation difficulties. The stress level at which the case will operate is 70,000 psi. Other approaches to the design of the motor hardware would perhaps work as well, but the above design was preferred for this application.

Purchased from American Institute of Aeronautics and Astronautics SOLID PROPELLANT ROCKET RESEARCH

MONOLITHIC GRAINS

The monolithic design, comprised of a single propellant grain bonded to the engine case wall, is the conventional approach to the design of high performance solid propellant motors. Since it is the simplest design from the standpoint of fabrication, it is the approach which will be discussed first. Because of certain problems characteristic of the motors considered, a grain supported with various inert hardware devices is considered as well as a design in which the grain is supported only at the case bond. The latter, casesupported monolithic design, will be considered first. CASE - SUPPORTED MONOLITHIC GRAIN

The problem of stresses induced in solid propellant grains by thermal, pressure, and inertia loadings constitutes one of the most severe limitations on the applications of solid propellants to large lightweight, high performance systems. The geometry used in the analysis of these problems is shown on Figure 2. The first loading considered is internal pressurization. Internal Pressurization The approach used in the analysis of the stresses and strains induced in the propellant grain is basically that derived by M. L. Williams and E. H. Lee. Expressions for the strain are determined using conventional elastic analysis. Then, the linearly viscoelastic properties of the propellant are substituted for the elastic constants in these expressions. Finally, the time-dependent loading is introduced, and the resulting equation is solved for the strain response. Elastic Analysis. The basic equations for the stresses in the grain were derived assuming the grain is an infinitely long, hollow circular cylinder, bonded at its outer periphery to an

elastic case, and unrestrained at the ends. The equations were modified in Reference (1) to account for stress concentration in the star points. The equations were then simplified, based on the recognition that the case stiffness is large compared to that of the grain. The critical stress was found to be the circumferential stress at the inside of the grain web. This stress is given by,

Purchased from American Institute of Aeronautics and Astronautics SOLID PROPELLANT ROCKET RESEARCH

P

~

where, G^LT - Radial stress at inside of grain web

°^i&- Circumferential stress at inside of grain web P = Internal pressure = 500 psi

K = Stress concentration factor at star root as defined in Reference (1) a = Inside web radius of grain - 54.0 in. b = Outside web radius of grain = 90.0 in. c = Outside radius of case

=

90.6 in.

/°= Radius of star root fillet =2.0 in. EC

=

Young's modulus of case = 9 x 10^ psi

E - Youngf s modulus of grain -\J = Poisson!s ratio of grain ^c6= Circumferential strain at inside of grain web A

= a/b

The tangential strain can be determined from Hook's law for plane stress as, Substituting equation (1) into equation (2) and noting that at the inside of the grain web 0r-r s - P:

r ^ EL

For the grain being analyzed,

A = a/b = 0.6O

Purchased from American Institute of Aeronautics and Astronautics SOLID PROPELLANT ROCKET RESEARCH

Assuming the propellant is imcompressible, Poisson's ratio, -J - .50. For the star point geometry,

^^ = 18,0

= O.t-O* From William's data (1), K = 3.50 Substituting A , K, and t^ into equation (3), This expression gives the strain the propellant would have to be capable of accommodating if it were an elastic material.

Actually, at temperatures above their glassy modulus temperature, propel lants are viscoelastic and therefore exhibit timedependent mechanical properties. If it is assumed that the propellant for the present design is linearly viscoelastic, the elastic solution given by equation (4) will represent the strain behavior of the material provided the time-dependent Young's modulus, E(t), of the propellant is substituted for E.

Viscoelastic Analysis. Assuming that the propellant1 s timedependent properties can be represented over a sufficiently long time range by the four-parameter model shown below, the material's properties will be given by,

where D = J/J t

Tl = ^ , /£", , A

————— Taking the Laplace transform of (5)

Equation (6) is the Laplace operator equation which represents the time dependent properties of the propellant. The next step is the determination of the model constants.

Purchased from American Institute of Aeronautics and Astronautics

SOLID PROPELLANT ROCKET RESEARCH

The materialf s stress relaxation modulus, E relaxation, representing the response of the material to a step increase in strain, Co , is given by the inverse Laplace transform of,

s Taking the inverse of (7),

ft 3)

Strain rate is given by,

u(t-t0)f ft-to)]

(it)

o( - 39.4 minutes The strain and strain rate given by equations (13) and (14) respectively are plotted on Figure 3. It should be recognized that these curves are based on the geometry at start of burning, and are therefore valid only for times before a significant amount of web has been burned away.

At the end of the pressure rise, before the strain rate has dropped, the strain is 29.2 per cent and the strain rate is 182 in./in-min. The cumulative effect of this strain together with the strain due to other loadings is discussed later in the paper.

Purchased from American Institute of Aeronautics and Astronautics SOLID PROPELLANT ROCKET RESEARCH

Slump

Axial loads induced by inertia of the propellant during acceleration of the rocket motor and by the weight of the propellant during storage of the motor on its end impose two requirements on the grain. The grain must be capable of carrying the induced axial shear stress, and it must resist excessive axial displacement. Shear Stress. The grain is assumed to be an infinitely long, hollow circular cylinder, supported in an upright position by a rigid case with no end restraint. The inside radius of the cylinder is assumed equal to the radius of the innermost point in the grain cross-section, a

The elastic shear stress in the grain due to an inertia force g times the force of gravity in the axial direction can be computed from,

The maximum shear stress occurs at the case bond (r - b), and for the present design is,

r = - 2, 5 8

sl

06)

Therefore the shear stress required of the material is seen to be quite low and should present no serious problem. Axial Displacement. The elastic axial displacement corresponding to the above shear stress is given by,

[2] where G = shear modulus Assuming an incompressible material (-\J~ .50) and G - E/3, the axial displacement at V = a. becomes,

ar = ^Brackets refer to references.

10

(17}

Purchased from American Institute of Aeronautics and Astronautics SOLID PROPELLANT ROCKET RESEARCH

Assuming that the propellant!s time-dependent creep properties can be represented over a sufficiently long time range by the four-parameter model shown below, the material!s Young's modulus will be given by, D 4 V

sN- S-i 5

P^S^-hPiS + PO

(20)

Substituting (20) into the elastic axial displacement equation (18), 5V ^-, 5

Assuming a step increase in g of gQ,

/5

(23)

Therefore,

11

Purchased from American Institute of Aeronautics and Astronautics

SOLID PROPELLANT ROCKET RESEARCH

Taking the inverse Laplace transform,

Substituting PQ, PI} q x , P2 into (25), / - -' t I U,. (t)= 3 27 j £./£,+>/£, - t/*z- ~Q^ J

(Z&)

This equation gives the timewise displacement response of the grain to a step increase in inertia loading. Since the corresponding shear stress, as given by equation (15), is directly proportional to "g" and is independent of the timedependent Young's modulus, a step increase in "g" results in a step increase in shear stress. Therefore, assuming the propellant is incompressible, the timewise displacement response of the grain to a step increase in "g", as given by equation (26), will have the same mode shape as the tensile creep response of a material specimen under constant stress loading. Tensile creep data for a typical propellant was fitted to a curve of the form of equation (26), and the elastic and viscous constants have been evaluated with the following results.

E-L - 5460 psi E3 = 3820 psi 0^2

= 18

'°

x 10

^ P si "

min

-

= 5454 psi - min. These constants are valid for times of 0.10 minute to 100. minutes after loading. Substituting in equation (26),

Ur fa - 3273oft